The Deligne-Beilinson conjecture predicts that the special values of many L-functions are related to the ranks of certain Ext groups in the category of mixed Hodge structures. In this talk, we present Skinner’s constructions of certain extensions that are extracted from the cohomology of the modular curve using CM points and the Eisenstein series. Through an explicit analytic calculation, which is performed in the adelic setting using (g,K)-cohomology and Tate’s zeta integrals, we obtain a formula relating the non-triviality of these extensions to the well-known non-vanishing at s=1 of the L-functions associated to Hecke characters of imaginary quadratic fields. These constructions have natural analogs in the category of p-adic Galois representations which are useful for Euler systems.

The superconformal index is one of the most powerful tools at the disposal of a supersymmetric field theorist. It counts protected states, is an RG flow invariant, and can be used to test for UV duality. Furthermore, it can be used to detect symmetry enhancements in the IR that are usually inaccessible by use of standard compactification or quiver techniques. The goal of this talk is to provide a practical introduction to computing indices. We will start with the supersymmetric harmonic oscillator to get some intuition, before building up a toolkit to compute indices for your favourite 4d N=1 SCFTs. Time permitting, we will discuss indices with N=2 supersymmetry.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

How many vectors are needed to simultaneously generate $m$ complete flags in $\mathbb{R}^d$, in the worst-case scenario?  A classical linear algebra fact, essentially equivalent to the Bruhat cell decomposition for $\text{GL}_d$, says that the answer is $d$ when $m=2$.  We obtain a precise answer for all values of $m$ and $d$.  Joint work with Federico Glaudo and Chayim Lowen.

 Hyper-Kähler manifolds are rigid geometric structures. They have three different symplectic and complex structures, in direct analogy with the quaternions. Being Ricci-flat, they solve the vacuum Einstein equations, and so there has been considerable interest among physicists to explicitly construct such spaces. We will discuss in detail the examples arising from the Gibbons-Hawking ansatz. These give concrete descriptions of the metric, giving many examples to work with. They also lead to the generalised classification as hyper-Kähler quotients by P.B. Kronheimer, with one such space for each finite subgroup of SU(2). Finally, we will look at the McKay correspondence, relating the finite subgroups of SU(2) with the simple Lie algebras of type A,D,E.